This report contains different plots and tables that may be relevant for analysing the results. Observe:
alg1Given a problem consisting of \(m\)
subproblems with \(Y_N^s\) given for
each subproblem \(s\), we use a
filtering algorithm to find \(Y_N\)
(alg1).
The following instance/problem groups are generated given:
u and l. [4 options]Note that the width of objective \(i = 1, \ldots p\), \(w_i = [l_i, u_i]\) should be approx. \(10000m\). Check:
## # A tibble: 4 × 6
## m mean_width1 mean_width2 mean_width3 mean_width4 mean_width5
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 2 19245. 19221. 19213. 18996. 18690.
## 2 3 28760. 28800. 28689. 28479. 27847.
## 3 4 38300. 38353. 38153. 37744. 36642.
## 4 5 47715. 47966. 47953. 47039. 44537.
What is \(|Y_N|\) given the different methods of generating the set of nondominated points for the subproblems?
## # A tibble: 4 × 3
## method mean_card n
## <chr> <dbl> <int>
## 1 l 325067. 235
## 2 m 419864. 305
## 3 u 102271. 305
## 4 ul 184743. 295
Does \(p\) have an effect?
## # A tibble: 16 × 4
## # Groups: method [4]
## method p mean_card n
## <chr> <dbl> <dbl> <int>
## 1 l 2 5829. 60
## 2 m 2 6828. 80
## 3 u 2 1164. 80
## 4 ul 2 2920. 80
## 5 l 3 61381. 60
## 6 m 3 180435. 80
## 7 u 3 12475. 80
## 8 ul 3 26863. 80
## 9 l 4 370968. 60
## 10 m 4 476291. 75
## 11 u 4 79341. 75
## 12 ul 4 185223. 75
## 13 l 5 910910 55
## 14 m 5 1105081. 70
## 15 u 5 345012. 70
## 16 ul 5 637081. 60
Does \(m\) have an effect?
## # A tibble: 15 × 4
## # Groups: method [4]
## method m mean_card n
## <chr> <dbl> <dbl> <int>
## 1 l 2 8173. 80
## 2 m 2 5688. 80
## 3 u 2 4201. 80
## 4 ul 2 4923. 80
## 5 l 3 166384. 80
## 6 m 3 90077. 80
## 7 u 3 37283. 80
## 8 ul 3 90425. 80
## 9 l 4 832349. 75
## 10 m 4 874692. 80
## 11 u 4 190675. 80
## 12 ul 4 485509. 80
## 13 m 5 775723. 65
## 14 u 5 194151. 65
## 15 ul 5 146013. 55
Let us try to fit the results using function \(y=c_1 s^{(c_2p)} m^{c_3p}\) (different functions was tried and this gave the highest \(R^2\)) for each method.
## # A tibble: 4 × 15
## method fit tidied r.squared adj.r.squared sigma statistic p.value df
## <chr> <list> <list> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 l <lm> <tibble> 0.804 0.802 1.04 475. 8.75e- 83 2
## 2 m <lm> <tibble> 0.765 0.764 1.22 492. 1.01e- 95 2
## 3 ul <lm> <tibble> 0.900 0.900 0.742 1317. 7.10e-147 2
## 4 u <lm> <tibble> 0.947 0.947 0.519 2705. 1.62e-193 2
## # ℹ 6 more variables: logLik <dbl>, AIC <dbl>, BIC <dbl>, deviance <dbl>,
## # df.residual <int>, nobs <int>
## # A tibble: 4 × 4
## method c1 c2 c3
## <chr> <dbl> <dbl> <dbl>
## 1 l 102. 0.0810 1.15
## 2 m 100. 0.0823 1.08
## 3 ul 31.6 0.117 1.10
## 4 u 24.5 0.134 0.946
We classify the nondominated points into, extreme, supported non-extreme and unsupported.
## # A tibble: 1 × 3
## minPctEx avePctExt maxPctEx
## <dbl> <dbl> <dbl>
## 1 0.000461 0.0469 0.330
## # A tibble: 4 × 4
## method minPctEx avePctExt maxPctEx
## <chr> <dbl> <dbl> <dbl>
## 1 l 0.0121 0.0929 0.302
## 2 ul 0.00639 0.0721 0.330
## 3 m 0.000461 0.0212 0.147
## 4 u 0.00196 0.0133 0.104